Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 22.8s

Alternatives: 17

Speedup: 0.5×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_0)) t_0))
      1.0)))))

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_0)) t_0))
      1.0)))))

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}}\\ t_1 := \left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - t\_0}\right) + \frac{1}{1 + t\_0}\\ \left(-s\right) \cdot \log \left(\frac{{t\_1}^{-3} + -1}{{t\_1}^{-2} + \left(1 + \frac{1}{t\_1}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ (PI) s)))
        (t_1
         (+
          (+ (/ u (+ 1.0 (exp (/ (PI) (- s))))) (/ u (- -1.0 t_0)))
          (/ 1.0 (+ 1.0 t_0)))))
   (*
    (- s)
    (log (/ (+ (pow t_1 -3.0) -1.0) (+ (pow t_1 -2.0) (+ 1.0 (/ 1.0 t_1))))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{{\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}} \cdot \left({\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - \left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-3} + -1}{{\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-2} + \left(1 + \frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)}\right)} \]

9. Final simplification 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{{\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-3} + -1}{{\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-2} + \left(1 + \frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)}\right) \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u + -1}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{-1 + {t\_0}^{-3}}{\frac{1}{t\_0} + \left(1 + {t\_0}^{-2}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0
         (+
          (/ u (+ 1.0 (exp (/ (PI) (- s)))))
          (/ (+ u -1.0) (- -1.0 (exp (/ (PI) s)))))))
   (*
    (- s)
    (log (/ (+ -1.0 (pow t_0 -3.0)) (+ (/ 1.0 t_0) (+ 1.0 (pow t_0 -2.0))))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{{\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}} \cdot \left({\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - \left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-3} + -1}{{\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-2} + \left(1 + \frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)}\right)} \]

10. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u - 1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-3} + -1}{\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u - 1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} + \left(1 + {\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u - 1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-2}\right)}\right)} \]

11. Final simplification 99.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u + -1}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-3}}{\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u + -1}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}} + \left(1 + {\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u + -1}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-2}\right)}\right) \]
12. Add Preprocessing

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}}\\ t_1 := \left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - t\_0}\right) + \frac{1}{1 + t\_0}\\ \left(-s\right) \cdot \log \left(\frac{-1 + {t\_1}^{-2}}{\frac{1}{t\_1} - -1}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ (PI) s)))
        (t_1
         (+
          (+ (/ u (+ 1.0 (exp (/ (PI) (- s))))) (/ u (- -1.0 t_0)))
          (/ 1.0 (+ 1.0 t_0)))))
   (* (- s) (log (/ (+ -1.0 (pow t_1 -2.0)) (- (/ 1.0 t_1) -1.0))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{{\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}} \cdot \left({\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - \left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-2} + -1}{\frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - -1}\right)} \]

9. Final simplification 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-2}}{\frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - -1}\right) \]
10. Add Preprocessing

Alternative 4: 14.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{1}{1 + t\_0} + u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{-1 - t\_0}\right)}\right) \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;s \cdot \frac{-1}{\frac{s}{\mathsf{PI}\left(\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ (PI) s))))
   (if (<=
        (*
         (- s)
         (log
          (+
           -1.0
           (/
            1.0
            (+
             (/ 1.0 (+ 1.0 t_0))
             (*
              u
              (+
               (/ 1.0 (+ 1.0 (exp (/ (PI) (- s)))))
               (/ 1.0 (- -1.0 t_0)))))))))
        -4.9999998413276127e-20)
     (* s (/ -1.0 (/ s (PI))))
     (/ (* s s) (* u (* (PI) 0.5))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -4.99999984e-20

   1. Initial program 98.8%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   4. Applied rewrites 98.7%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

   5. Taylor expanded in u around 0

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
   6. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

      2. lower-PI.f32 15.2

         \[\leadsto \left(-s\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

   7. Applied rewrites 15.2%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
   8. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

      2. clear-num N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{1}{\frac{s}{\mathsf{PI}\left(\right)}}} \]

      3. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{1}{\frac{s}{\mathsf{PI}\left(\right)}}} \]

      4. lower-/.f32 15.2

         \[\leadsto \left(-s\right) \cdot \frac{1}{\color{blue}{\frac{s}{\mathsf{PI}\left(\right)}}} \]

   9. Applied rewrites 15.2%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{1}{\frac{s}{\mathsf{PI}\left(\right)}}} \]

   if -4.99999984e-20 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 99.2%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log -1\right) + \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(s \cdot \log -1\right)\right)} + \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{s \cdot \left(\mathsf{neg}\left(\log -1\right)\right)} + \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right)} \]

      4. lower-neg.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \color{blue}{\mathsf{neg}\left(\log -1\right)}, \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right) \]

      5. lower-log.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\color{blue}{\log -1}\right), \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \color{blue}{\frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}}\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}}\right) \]

      9. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}}\right) \]

   5. Applied rewrites 10.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(s, -\log -1, \frac{s}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right)} \]

   6. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{{s}^{2}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{s}^{2}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{s \cdot s}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{s \cdot s}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{s \cdot s}{\color{blue}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \frac{s \cdot s}{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{s \cdot s}{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}} \]

      8. lower-PI.f32 13.8

         \[\leadsto \frac{s \cdot s}{u \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right)} \]

   8. Applied rewrites 13.8%

      \[\leadsto \color{blue}{\frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right) \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;s \cdot \frac{-1}{\frac{s}{\mathsf{PI}\left(\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 14.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{1}{1 + t\_0} + u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{-1 - t\_0}\right)}\right) \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \left(s \cdot \frac{-1}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ (PI) s))))
   (if (<=
        (*
         (- s)
         (log
          (+
           -1.0
           (/
            1.0
            (+
             (/ 1.0 (+ 1.0 t_0))
             (*
              u
              (+
               (/ 1.0 (+ 1.0 (exp (/ (PI) (- s)))))
               (/ 1.0 (- -1.0 t_0)))))))))
        -4.9999998413276127e-20)
     (* (PI) (* s (/ -1.0 s)))
     (/ (* s s) (* u (* (PI) 0.5))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -4.99999984e-20

   1. Initial program 98.8%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   4. Applied rewrites 98.7%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

   5. Taylor expanded in u around 0

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
   6. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

      2. lower-PI.f32 15.2

         \[\leadsto \left(-s\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

   7. Applied rewrites 15.2%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
   8. Step-by-step derivation

      1. lift-neg.f32 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

      2. lift-PI.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

      3. lift-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{s} \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

      5. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \cdot \left(\mathsf{neg}\left(s\right)\right) \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{s}\right)} \cdot \left(\mathsf{neg}\left(s\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{s} \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{s} \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{s} \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \]

      10. lower-/.f32 15.2

          \[\leadsto \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\frac{1}{s}} \cdot \left(-s\right)\right) \]

   9. Applied rewrites 15.2%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{s} \cdot \left(-s\right)\right)} \]

   if -4.99999984e-20 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 99.2%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log -1\right) + \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(s \cdot \log -1\right)\right)} + \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{s \cdot \left(\mathsf{neg}\left(\log -1\right)\right)} + \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right)} \]

      4. lower-neg.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \color{blue}{\mathsf{neg}\left(\log -1\right)}, \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right) \]

      5. lower-log.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\color{blue}{\log -1}\right), \frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \color{blue}{\frac{s}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}}\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}}\right) \]

      9. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, \mathsf{neg}\left(\log -1\right), \frac{s}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}}\right) \]

   5. Applied rewrites 10.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(s, -\log -1, \frac{s}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right)} \]

   6. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{{s}^{2}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{s}^{2}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{s \cdot s}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{s \cdot s}}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{s \cdot s}{\color{blue}{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \frac{s \cdot s}{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{s \cdot s}{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}} \]

      8. lower-PI.f32 13.8

         \[\leadsto \frac{s \cdot s}{u \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right)} \]

   8. Applied rewrites 13.8%

      \[\leadsto \color{blue}{\frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}\right) \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \left(s \cdot \frac{-1}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}}\\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - t\_0}\right) + \frac{1}{1 + t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ (PI) s))))
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       1.0
       (+
        (+ (/ u (+ 1.0 (exp (/ (PI) (- s))))) (/ u (- -1.0 t_0)))
        (/ 1.0 (+ 1.0 t_0)))))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{{\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}} \cdot \left({\left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - \left(\frac{-u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} - \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

9. Final simplification 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{1}{\left(\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{u}{-1 - e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right) \]
10. Add Preprocessing

Alternative 7: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ (PI) (- s)))))
      (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
4. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right) + \frac{\mathsf{PI}\left(\right)}{s}\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{\left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + 1\right)}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\color{blue}{\frac{\frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\frac{\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{2}}}{{s}^{2}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{\frac{1}{2}}{{s}^{2}}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   13. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   14. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   15. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\frac{1}{2}}{{s}^{2}}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   16. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{\color{blue}{s \cdot s}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   17. lower-*.f32 95.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{\color{blue}{s \cdot s}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

5. Applied rewrites 95.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

6. Taylor expanded in s around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. lower-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. lower-neg.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. lower-PI.f32 98.7

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\color{blue}{\mathsf{PI}\left(\right)}}{s}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

8. Applied rewrites 98.7%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

9. Final simplification 98.7%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right) \]
10. Add Preprocessing

Alternative 8: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{-1 - \left(t\_0 + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + \left(1 + t\_0\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       1.0
       (+
        (*
         u
         (+
          (/ 1.0 (+ 1.0 (exp (/ (PI) (- s)))))
          (/ 1.0 (- -1.0 (+ t_0 (fma (* (PI) (PI)) (/ 0.5 (* s s)) 1.0))))))
        (/ 1.0 (+ 1.0 (+ 1.0 t_0))))))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
4. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right) + \frac{\mathsf{PI}\left(\right)}{s}\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{\left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + 1\right)}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\color{blue}{\frac{\frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\frac{\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{2}}}{{s}^{2}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{\frac{1}{2}}{{s}^{2}}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   13. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   14. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   15. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\frac{1}{2}}{{s}^{2}}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   16. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{\color{blue}{s \cdot s}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   17. lower-*.f32 95.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{\color{blue}{s \cdot s}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

5. Applied rewrites 95.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

6. Taylor expanded in s around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}} - 1\right) \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}} - 1\right) \]

   2. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + \left(1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right)}} - 1\right) \]

   3. lower-PI.f32 86.1

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + \left(1 + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}\right)}} - 1\right) \]

8. Applied rewrites 86.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}} - 1\right) \]

9. Final simplification 86.1%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{1}{-1 - \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}\right) \]
10. Add Preprocessing

Alternative 9: 37.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{1}{1 + e^{t\_0}} + u \cdot \left(\frac{1}{1 + 1} + \frac{1}{-1 - \left(t\_0 + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       1.0
       (+
        (/ 1.0 (+ 1.0 (exp t_0)))
        (*
         u
         (+
          (/ 1.0 (+ 1.0 1.0))
          (/
           1.0
           (- -1.0 (+ t_0 (fma (* (PI) (PI)) (/ 0.5 (* s s)) 1.0)))))))))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
4. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right) + \frac{\mathsf{PI}\left(\right)}{s}\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} + \left(1 + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{\left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + 1\right)}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\color{blue}{\frac{\frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\frac{\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{2}}}{{s}^{2}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{\frac{1}{2}}{{s}^{2}}} + 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   13. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   14. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{2}}{{s}^{2}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   15. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\frac{1}{2}}{{s}^{2}}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   16. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{\color{blue}{s \cdot s}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   17. lower-*.f32 95.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{\color{blue}{s \cdot s}}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

5. Applied rewrites 95.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

6. Taylor expanded in s around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{1}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2}}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
7. Step-by-step derivation

8. Applied rewrites 37.7%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{1}} - \frac{1}{1 + \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

9. Final simplification 37.7%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + u \cdot \left(\frac{1}{1 + 1} + \frac{1}{-1 - \left(\frac{\mathsf{PI}\left(\right)}{s} + \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5}{s \cdot s}, 1\right)\right)}\right)}\right) \]
10. Add Preprocessing

Alternative 10: 3.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(0.25, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{u}\right)} \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \mathsf{PI}\left(\right) \cdot 0.25\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (*
     (/ s (* (* u u) (fma 0.25 (* (PI) (PI)) (/ (* 0.5 (* s (PI))) u))))
     (fma 0.5 (fma u (PI) s) (* (PI) 0.25)))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}}} - 1\right) \]
4. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}}} - 1\right) \]

   2. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{1}{2} + \color{blue}{\frac{-1 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}}} - 1\right) \]

5. Applied rewrites 0.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{0.5 + \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.25, u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)\right)}{s}}} - 1\right) \]

6. Taylor expanded in s around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{2} \cdot s + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}{s}}} - 1\right) \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{2} \cdot s + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}{s}}} - 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(\frac{1}{2} \cdot s + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{s}} - 1\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\frac{1}{2} \cdot \left(s + u \cdot \mathsf{PI}\left(\right)\right)} + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}} - 1\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, s + u \cdot \mathsf{PI}\left(\right), \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}}{s}} - 1\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{s + u \cdot \mathsf{PI}\left(\right)}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \color{blue}{\mathsf{PI}\left(\right) \cdot u}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \color{blue}{\mathsf{PI}\left(\right) \cdot u}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   8. lower-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \color{blue}{\mathsf{PI}\left(\right)} \cdot u, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right)}{s}} - 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right)}{s}} - 1\right) \]

   11. lower-PI.f32 -0.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(0.5, s + \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{PI}\left(\right)} \cdot -0.25\right)}{s}} - 1\right) \]

8. Applied rewrites -0.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, s + \mathsf{PI}\left(\right) \cdot u, \mathsf{PI}\left(\right) \cdot -0.25\right)}{s}}} - 1\right) \]

10. Applied rewrites 0.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{s}{\mathsf{fma}\left(0.25, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right) \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), -\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.0625\right)} \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), 0.25 \cdot \mathsf{PI}\left(\right)\right)} - 1\right) \]

11. Taylor expanded in u around inf

    \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\color{blue}{{u}^{2} \cdot \left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)}} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]
12. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\color{blue}{{u}^{2} \cdot \left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)}} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\color{blue}{\left(u \cdot u\right)} \cdot \left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\color{blue}{\left(u \cdot u\right)} \cdot \left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    4. lower-fma.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)}} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    5. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    6. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    7. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    8. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    9. associate-*r/ N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\frac{1}{2} \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{u}}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    10. lower-/.f32 N/A

        \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\frac{1}{2} \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{u}}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    11. lower-*.f32 N/A

        \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\frac{1}{2} \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    12. lower-*.f32 N/A

        \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\frac{1}{2} \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}}{u}\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

    13. lower-PI.f32 18.5

        \[\leadsto \left(-s\right) \cdot \log \left(\frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(0.25, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5 \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{u}\right)} \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), 0.25 \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

13. Applied rewrites 18.5%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{s}{\color{blue}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(0.25, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{u}\right)}} \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), 0.25 \cdot \mathsf{PI}\left(\right)\right) - 1\right) \]

14. Final simplification 18.5%

    \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{s}{\left(u \cdot u\right) \cdot \mathsf{fma}\left(0.25, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{0.5 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{u}\right)} \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), \mathsf{PI}\left(\right) \cdot 0.25\right)\right) \]
15. Add Preprocessing

Alternative 11: 23.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \left(\mathsf{fma}\left(-8, \frac{\mathsf{fma}\left(0.5, u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right), \mathsf{PI}\left(\right) \cdot 0.25\right)}{s}, 2\right) + 0.25 \cdot \frac{\mathsf{PI}\left(\right) \cdot 12}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (+
     (fma -8.0 (/ (fma 0.5 (* u (* (PI) 0.5)) (* (PI) 0.25)) s) 2.0)
     (* 0.25 (/ (* (PI) 12.0) s)))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

5. Taylor expanded in s around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{\left(\left(2 + -8 \cdot \frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{s}\right) - \frac{-1}{4} \cdot \frac{4 \cdot \mathsf{PI}\left(\right) + 8 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{\left(\left(2 + -8 \cdot \frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{s}\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{4 \cdot \mathsf{PI}\left(\right) + 8 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

   2. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{\left(\left(2 + -8 \cdot \frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{s}\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{4 \cdot \mathsf{PI}\left(\right) + 8 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

7. Applied rewrites 20.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(\mathsf{fma}\left(-8, \frac{\mathsf{fma}\left(0.5, u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right), \mathsf{PI}\left(\right) \cdot 0.25\right)}{s}, 2\right) + 0.25 \cdot \frac{\mathsf{PI}\left(\right) \cdot 12}{s}\right)} - 1\right) \]

8. Final simplification 20.3%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \left(\mathsf{fma}\left(-8, \frac{\mathsf{fma}\left(0.5, u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right), \mathsf{PI}\left(\right) \cdot 0.25\right)}{s}, 2\right) + 0.25 \cdot \frac{\mathsf{PI}\left(\right) \cdot 12}{s}\right)\right) \]
9. Add Preprocessing

Alternative 12: 12.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + \mathsf{fma}\left(0.5, \frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot -4}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (+ 1.0 (fma 0.5 (/ (* (* u (PI)) -4.0) s) (/ (PI) s))))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}}} - 1\right) \]
4. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{2} + -1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}}} - 1\right) \]

   2. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{1}{2} + \color{blue}{\frac{-1 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}}} - 1\right) \]

5. Applied rewrites 0.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{0.5 + \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.25, u \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)\right)}{s}}} - 1\right) \]

6. Taylor expanded in s around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{2} \cdot s + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}{s}}} - 1\right) \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{2} \cdot s + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}{s}}} - 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(\frac{1}{2} \cdot s + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{s}} - 1\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\frac{1}{2} \cdot \left(s + u \cdot \mathsf{PI}\left(\right)\right)} + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}} - 1\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, s + u \cdot \mathsf{PI}\left(\right), \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}}{s}} - 1\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{s + u \cdot \mathsf{PI}\left(\right)}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \color{blue}{\mathsf{PI}\left(\right) \cdot u}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \color{blue}{\mathsf{PI}\left(\right) \cdot u}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   8. lower-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \color{blue}{\mathsf{PI}\left(\right)} \cdot u, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - 1\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right)}{s}} - 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, s + \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right)}{s}} - 1\right) \]

   11. lower-PI.f32 -0.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(0.5, s + \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{PI}\left(\right)} \cdot -0.25\right)}{s}} - 1\right) \]

8. Applied rewrites -0.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, s + \mathsf{PI}\left(\right) \cdot u, \mathsf{PI}\left(\right) \cdot -0.25\right)}{s}}} - 1\right) \]

10. Applied rewrites 0.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{s}{\mathsf{fma}\left(0.25, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right) \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), -\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.0625\right)} \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, \mathsf{PI}\left(\right), s\right), 0.25 \cdot \mathsf{PI}\left(\right)\right)} - 1\right) \]

11. Taylor expanded in s around inf

    \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{4 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - 8 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{s} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)} \]
12. Step-by-step derivation

    1. lower-+.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{4 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - 8 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{s} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)} \]

    2. lower-fma.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{4 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - 8 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]

    3. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{4 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - 8 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{s}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    4. distribute-rgt-out-- N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \left(4 - 8\right)}}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \mathsf{fma}\left(\frac{1}{2}, \frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{-4}}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    6. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot -4}}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    7. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right)} \cdot -4}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    8. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \mathsf{fma}\left(\frac{1}{2}, \frac{\left(u \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot -4}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    9. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 + \mathsf{fma}\left(\frac{1}{2}, \frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot -4}{s}, \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right) \]

    10. lower-PI.f32 16.7

        \[\leadsto \left(-s\right) \cdot \log \left(1 + \mathsf{fma}\left(0.5, \frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot -4}{s}, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}\right)\right) \]

13. Applied rewrites 16.8%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + \mathsf{fma}\left(0.5, \frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot -4}{s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)} \]
14. Add Preprocessing

Alternative 13: 14.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ \frac{\frac{t\_0 \cdot t\_0}{\left(-s\right) \cdot \mathsf{PI}\left(\right)}}{s} \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* s (PI)))) (/ (/ (* t_0 t_0) (* (- s) (PI))) s)))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

5. Taylor expanded in u around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

   2. lower-PI.f32 11.3

      \[\leadsto \left(-s\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

7. Applied rewrites 11.3%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
8. Step-by-step derivation

   1. lift-neg.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   2. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{PI}\left(\right)}{s}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{PI}\left(\right)}{s}} \]

   5. lift-neg.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \mathsf{PI}\left(\right)}{s} \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(s \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   7. lower-neg.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(s \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   8. lower-*.f32 11.3

      \[\leadsto \frac{-\color{blue}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

9. Applied rewrites 11.3%

   \[\leadsto \color{blue}{\frac{-s \cdot \mathsf{PI}\left(\right)}{s}} \]
10. Step-by-step derivation

    1. lift-PI.f32 N/A

       \[\leadsto \frac{\mathsf{neg}\left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{s} \]

    2. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

    3. neg-sub0 N/A

       \[\leadsto \frac{\color{blue}{0 - s \cdot \mathsf{PI}\left(\right)}}{s} \]

    4. flip-- N/A

       \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{0 + s \cdot \mathsf{PI}\left(\right)}}}{s} \]

    5. lower-/.f32 N/A

       \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{0 + s \cdot \mathsf{PI}\left(\right)}}}{s} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{0} - \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{0 + s \cdot \mathsf{PI}\left(\right)}}{s} \]

    7. lower--.f32 N/A

       \[\leadsto \frac{\frac{\color{blue}{0 - \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}{0 + s \cdot \mathsf{PI}\left(\right)}}{s} \]

    8. lower-*.f32 N/A

       \[\leadsto \frac{\frac{0 - \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}{0 + s \cdot \mathsf{PI}\left(\right)}}{s} \]

    9. lower-+.f32 14.3

       \[\leadsto \frac{\frac{0 - \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{0 + s \cdot \mathsf{PI}\left(\right)}}}{s} \]

11. Applied rewrites 14.3%

    \[\leadsto \frac{\color{blue}{\frac{0 - \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{0 + s \cdot \mathsf{PI}\left(\right)}}}{s} \]

12. Final simplification 14.3%

    \[\leadsto \frac{\frac{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot \mathsf{PI}\left(\right)}}{s} \]
13. Add Preprocessing

Alternative 14: 13.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{PI}\left(\right)}{s} \cdot \left(\left(s \cdot s\right) \cdot \frac{-1}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (/ (PI) s) (* (* s s) (/ -1.0 s))))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

5. Taylor expanded in u around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

   2. lower-PI.f32 11.3

      \[\leadsto \left(-s\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

7. Applied rewrites 11.3%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

9. Applied rewrites 14.0%

   \[\leadsto \color{blue}{\left(\left(s \cdot s\right) \cdot \frac{-1}{s}\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

10. Final simplification 14.0%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{s} \cdot \left(\left(s \cdot s\right) \cdot \frac{-1}{s}\right) \]
11. Add Preprocessing

Alternative 15: 13.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{PI}\left(\right)}{-s} \cdot \frac{s \cdot s}{s} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (/ (PI) (- s)) (/ (* s s) s)))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

5. Taylor expanded in u around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

   2. lower-PI.f32 11.3

      \[\leadsto \left(-s\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

7. Applied rewrites 11.3%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
8. Step-by-step derivation

   1. lift-neg.f32 11.3

      \[\leadsto \color{blue}{\left(-s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   2. --rgt-identity N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) - 0\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   3. flip-- N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot 0}{\left(\mathsf{neg}\left(s\right)\right) + 0}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} - 0 \cdot 0}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - \color{blue}{0}}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   6. --rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)}}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)}{\left(\mathsf{neg}\left(s\right)\right) + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   8. sub-neg N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(s\right)\right) - 0}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   9. --rgt-identity N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)}{\color{blue}{\mathsf{neg}\left(s\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   10. lower-/.f32 14.0

       \[\leadsto \color{blue}{\frac{\left(-s\right) \cdot \left(-s\right)}{-s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)}}{\mathsf{neg}\left(s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   12. lift-neg.f32 N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \left(\mathsf{neg}\left(s\right)\right)}{\mathsf{neg}\left(s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   13. lift-neg.f32 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}}{\mathsf{neg}\left(s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   14. sqr-neg N/A

       \[\leadsto \frac{\color{blue}{s \cdot s}}{\mathsf{neg}\left(s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   15. lift-*.f32 14.0

       \[\leadsto \frac{\color{blue}{s \cdot s}}{-s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

9. Applied rewrites 14.0%

   \[\leadsto \color{blue}{\frac{s \cdot s}{-s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

10. Final simplification 14.0%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{-s} \cdot \frac{s \cdot s}{s} \]
11. Add Preprocessing

Alternative 16: 11.4% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \frac{s \cdot \mathsf{PI}\left(\right)}{-s} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (/ (* s (PI)) (- s)))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{2} + \frac{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} - u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{-s}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)}^{3} + {\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}^{-3}}}}} - 1\right) \]

5. Taylor expanded in u around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

   2. lower-PI.f32 11.3

      \[\leadsto \left(-s\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

7. Applied rewrites 11.3%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
8. Step-by-step derivation

   1. lift-neg.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   2. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{s} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{PI}\left(\right)}{s}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{PI}\left(\right)}{s}} \]

   5. lift-neg.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \mathsf{PI}\left(\right)}{s} \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(s \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   7. lower-neg.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(s \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   8. lower-*.f32 11.3

      \[\leadsto \frac{-\color{blue}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

9. Applied rewrites 11.3%

   \[\leadsto \color{blue}{\frac{-s \cdot \mathsf{PI}\left(\right)}{s}} \]

10. Final simplification 11.3%

    \[\leadsto \frac{s \cdot \mathsf{PI}\left(\right)}{-s} \]
11. Add Preprocessing

Alternative 17: 11.4% accurate, 170.0× speedup

Math

\[\begin{array}{l} \\ -\mathsf{PI}\left(\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- (PI)))

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   3. lower-PI.f32 11.3

      \[\leadsto -\color{blue}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 11.3%

   \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) (/ 1.0 (+ 1.0 (exp (/ (PI) s)))))) (/ 1.0 (+ 1.0 (exp (/ (PI) s)))))) 1.0))))